Mathematics 0580 · syllabus C1.1, C1.5, C1.6, E1.1, E1.5, E1.6
Cambridge IGCSE Mathematics 0580 — Types of number and the four operations topical past papers
Primes, factors, multiples, HCF and LCM, reciprocals and ordering, with calculation by hand using negative numbers, fractions and mixed numbers.
- Questions
- 1,155
- Years
- 2009–2026
- With mark scheme
- 90%
- Questions 2022–2026
- 415
Anyone can browse these pages. Opening the questions and their mark schemes takes a free Momentum account.
What Types of number and the four operations covers
This topic is the vocabulary and arithmetic that the rest of the course takes for granted. You need to recognise natural numbers, integers, primes, square and cube numbers, factors, multiples and reciprocals, and to write numbers in words and in figures. A typical task is writing a number as a product of its prime factors, such as 360 = 2³ × 3² × 5, and using that form to work out the HCF (highest common factor) and LCM (lowest common multiple) of a pair of numbers. You also order quantities and use the symbols =, ≠, <, >, ≤ and ≥ correctly.
The four operations are calculation by hand: adding, subtracting, multiplying and dividing integers, decimals and fractions, including negative numbers, improper fractions and mixed numbers, in the right order and with brackets where they are needed. Practical settings, such as a temperature that falls below zero, are part of the topic. Since the current syllabus made Papers 1 and 2 non-calculator papers, these skills earn marks on their own and decide whether longer questions on other topics end without a slip.
Everything here is Core content, so it can be set on all four papers. Prime factors lead on to indices and to simplifying surds, fractions to the conversions between fractions, decimals and percentages, and the inequality symbols to inequalities. Telling rational numbers from irrational ones belongs to the same syllabus point and has a chapter of its own, with surds.
In the syllabus
The headings this topic has in the Cambridge International 2025–2027 syllabus, as the board prints them.
- C1.1 Types of number
- C1.5 Ordering
- C1.6 The four operations
- E1.1 Types of number
- E1.5 Ordering
- E1.6 The four operations
What you should be able to do
- Pick out primes, squares, cubes, factors, multiples and reciprocals from a list of numbers
- Write a number as a product of prime factors in index form
- Find the HCF and LCM of two or three numbers, and use them in problems
- Order integers, decimals and fractions, using the symbols =, ≠, <, >, ≤ and ≥
- Add, subtract, multiply and divide fractions and mixed numbers by hand
- Calculate with negative numbers and follow the order of operations, including brackets
Key ideas
- A prime has exactly two factors, 1 and itself, so 1 is not prime and 2 is the only even prime
- HCF: multiply the primes the numbers share, each to the lower power; LCM: multiply every prime that appears, each to the higher power
- For two positive whole numbers, HCF × LCM = the product of the numbers
- The reciprocal of x is 1/x, so the reciprocal of 2/5 is 5/2, and a number times its reciprocal is 1
- Order of operations: brackets, then powers, then multiplication and division from left to right, then addition and subtraction from left to right
How Types of number and the four operations has been examined
From 2022 to 2026, Types of number and the four operations was on Paper 3 · Calculator (Core) in 14 of 14 exam series, worth about 18% of that paper’s marks.
| Paper | Questions | Share of marks | Series |
|---|---|---|---|
| Paper 1 · Non-calculator (Core) | 208 | 22% | 14 of 14 |
| Paper 2 · Non-calculator (Extended) | 107 | 9% | 14 of 14 |
| Paper 3 · Calculator (Core) | 86 | 18% | 14 of 14 |
| Paper 4 · Calculator (Extended) | 14 | 2% | 8 of 14 |
Momentum’s own count, over the questions in its bank for these years: a series is one exam session, every zone included, and a question filed under two chapters counts in both. Median marks per question: 2.
Filed in Momentum under: Number Facts.
A worked example
Write 252 and 360 as products of their prime factors. Hence find the highest common factor and the lowest common multiple of 252 and 360.
- 252 = 2 × 126 = 2 × 2 × 63 = 2 × 2 × 3 × 21 = 2 × 2 × 3 × 3 × 7, so 252 = 2² × 3² × 7.
- 360 = 2 × 180 = 2 × 2 × 90 = 2 × 2 × 2 × 45 = 2 × 2 × 2 × 3 × 3 × 5, so 360 = 2³ × 3² × 5.
- HCF: take the primes in both numbers, 2 and 3, each to its lower power: 2² × 3² = 4 × 9 = 36.
- LCM: take every prime that appears, each to its higher power: 2³ × 3² × 5 × 7 = 8 × 9 × 35 = 2520.
- Check: 36 × 2520 = 90 720, which equals 252 × 360.
Answer: 252 = 2² × 3² × 7 and 360 = 2³ × 3² × 5; HCF = 36 and LCM = 2520
Written by Momentum for this page, with its own numbers; it is not taken from a past paper.
Mistakes that cost marks
- Counting 1 as a prime: a prime has exactly two factors and 1 has only one, so the primes start at 2.
- Mixing up the HCF and the LCM: the HCF can be no larger than the smaller number and the LCM no smaller than the larger, so a quick size check catches the swap.
- Adding fractions by adding the tops and the bottoms: 1/3 + 1/4 is 7/12, not 2/7, because the fractions need a common denominator first.
- Working strictly from left to right: 6 + 4 × 3 is 18, not 30, because multiplication comes before addition.
- Subtracting a negative number as if it were positive: 3 − (−5) = 8, not −2.
- Leaving an improper fraction when the answer is wanted as a mixed number in lowest terms: the final mark is for the form as well as the value.
Exam technique
- Use a factor tree or repeated division by primes, write the result in index form, and check that it multiplies back to the original number.
- For fraction calculations, change mixed numbers to improper fractions, cancel before multiplying, and write every line: on the non-calculator papers the working earns method marks.
- Check each answer by estimating or by reversing the operation; it takes seconds and catches sign and place-value slips.
Practising in Momentum
- Every question in a chapter as one list, from every paper and year in the bank, newest first or shuffled.
- Each question turns over to its own part of the mark scheme, where the board published one.
- Narrow by paper, year, session, zone and marks, and star the questions worth keeping.
- Pick chapters, a number of questions and a time, and the exam builder draws a mock to sit and mark.
Past-paper questions and mark schemes are the copyright of Cambridge International. Momentum shows them to signed-in learners for revision and does not publish them on these pages.